Leftover vertices after parenting to bone - mouth/eyeballs not rigged
You often need to make some corrections once you've parented, typically with independent meshes like eyes, button, etc...

The classic way to fix this is to switch your armature to Pose mode and switch your object to Weight Paint mode then to select the bone to which you want to assign the button/eye, and paint this zone with the brush. Don't forget to deactivate the Limit Selection To Visible option, button on the right of the header menu.



You can also assign the vertices you want to the vertex group they are supposed to be part of in the Properties panel > Data > Vertex Group... When you've parented With Automatic Weights it has automatically created vertex groups and given them the name of the bone that controls them. If you want to add some vertices to a group, select those vertices in your mesh, select the group in the list and press the Assign button.